Lab 4:The derivative of y = ex.Math 131 AB2/14/06
Part #1: Exploring the derivative of y = ax numerically and graphically.
We will explore y = f(x) = 2x and y = g(x) =3x along with f '(x) and g '(x) to conjecture the derivative of y = ex.
a) Construct data tables for numerical comparisons (set Mode:Display Digits to FIX 4) for y = f(x) =2x. On your calculator enter:
y1(x)=2^x and then use the TABLE function:seq(2^x, x, 1.0, 1.5, 0.1)
y2(x)=d(y1(x),x) [APPS][5:Table] , [F2 Setup] orseq(d(2^x, x), x, 1.0, 1.5, 0.1)
y3(x)=y2(x)/y1(x) with TblStart = 1 and Tbl =.1seq((d(2^x, x)/ 2^x), x, 1.0, 1.5, 0.1)
i) What is the significance of the third function in the table? (Note: )
ii.) Use Graph to sketch these functions on the same axes. Start with the window [-2, 4]x and [-1, 15]y.
iii.) Try to write a formula for . (Hint: find the ln 2.)
b) Repeat (a) for the function y = g(x) = 3x, but use y4, y5, & y6 as storage locations.(Hint: find the ln 3.)
c) Try to answer this question: For what value of a will the graph of y = ax and its derivative coincide? Repeat (a) for that value of a, but use y7, y8, & y9 as storage locations.
d) Sketch y1, y2, y4, y5, y7, & y8 on the same set of axes. Comment on what you see.
Part #2: Computing the derivative of y = ex algebraically.
a) We will need to evaluate .
i) First evaluate the function for values of h approaching 0 & use the tables to estimate the limit.
ii) Use the graph of the function to estimate the limit. Show the graph of the function and the window range you selected and explain how you obtained the result. Comment on the behavior of the function at h=0 and its graph as depicted on your calculator.
b)Use the result from (a)and the difference quotient to find .
Part #3: Considering the graphs.
a) How does the derivative of y = ax where a > 1 resemble the function graphically?
b) Sketch the graph of
Compute its derivative. Consider the sign of its derivative and the graph of the derivative.
Discuss how the derivatives of y = ax where 0 < a < 1 resemble the function. Do they have the same “shape” as the function as when a > 1?
Part #4: See text page 121 Work #41.